{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra, Plots;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PortfolioChoice (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function PortfolioChoice(saving ;b_e_bar=0.21, ϕ=0.05, r_d=0.02/4, r_e=0.02/4)\n",
    "    #= \n",
    "    Min: q_d*b_d + q_e*b_e + Φ(b_e)\n",
    "    s.t: b_d + b_e = b\n",
    "    =>\n",
    "    q_d = q_e + Φ′(b_e)\n",
    "    under the assumption Φ(b_e)=ϕ/2*(b_e - b_e_bar)^2\n",
    "    =#\n",
    "    q_d = 1/(1+r_d); q_e = 1/(1+r_e);\n",
    "    b_e = (q_d-q_e)/ϕ + b_e_bar;\n",
    "    b_d = (saving .- q_e*b_e .- ϕ/2*(b_e-b_e_bar)^2)/q_d;\n",
    "    b_e = length(saving)>1 ? fill(b_e, size(saving)) : b_e;\n",
    "    EffRet = (b_d + b_e) ./ saving .- 1;\n",
    "    b = b_d + b_e;\n",
    "    return b_e, b_d, EffRet, b\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "saving = range(-1, 5, length=50);\n",
    "temp_0 = PortfolioChoice(saving);\n",
    "temp_1 = PortfolioChoice(saving; r_e=(0.02+0.02)/4);\n",
    "temp_2 = PortfolioChoice(saving; r_e=(0.02+0.04)/4);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(saving, temp_1[3] .- 0.02/4)\n",
    "plot!(saving, temp_2[3] .- 0.02/4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "List_ϕ = range(0, 1, length=50);\n",
    "ret_pos = [PortfolioChoice(1 ;b_e_bar=0.21, ϕ=xx, r_d=0.02/4, r_e=0.04/4)[3]-0.02/4 for xx in List_ϕ];\n",
    "ret_neg = [PortfolioChoice(-0.1 ;b_e_bar=0.21, ϕ=xx, r_d=0.02/4, r_e=0.04/4)[3]-0.02/4 for xx in List_ϕ];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(List_ϕ, ret_pos, label=\"Positive saving\")\n",
    "plot!(List_ϕ, ret_neg, label=\"Negative saving\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AseqAhUYBwKDaFIR5eXlXXHHFI488EgwGi4uL33///Wuuuaa+vv6nP/3p0aNHZ8+e/eijj7rd7mPHji1duvT666/Py8ubOHFi6PxNmzYVFxfPmjVr0qRJ5eXl//znP0VkyZIlffr0Offcc+N0dziJDkIAaElbZwU/9thjW7ZsycnJufLKK//617926dKloaHh3//+d01Nza233tqnT5+uXbv269dv5syZU6dODZ2/efPmnJycadOmLV26tEuXLg6H48UXX/z5z3+elZW1bNmy5cuXx+O+cApPQHNYk60DCQDiok3TJ8J8Pl9LHTN+v99isTTpq2z2/AgXCYl1+kRJea3TlZbrMOywudY6ffrEnmptwuuB3TOSvDvBzH2ETBYyGzMPlnG73fHtI4zPLzHCR0+zf6dmz4/759dfvrQ+s7uhwiedUiTXoXRySG6KkuuQvFTJdSidUqTQpRQ4pZtLOX0ucJKhaRQAWpLM3ybuHex/eFRqQJMyj5R7tTKPlHu0Yx4p88jXtdqHZXLYHTzilgN1mqZJ9zSla6p0cykFLilwKt1c0jNN6ZeZJBnJYBkAaEkyB2GIVZEuqdIlNdQ+2nwraW2DHKjTjrjlkFs7VCe7q7UNR2RvdfCrai3DJmdnKn0zlH6ZSr8M6Zep9MtQUo32a6NGCAAtMdonevtIs8mALGVAlpyelAfrtK+qZXeV9lW1tmK3fFUd3FOtdXYog3Pkgs6WEZ2VC/KUnDN+3TJv8IzegwkAEoggjKKbS+nmku92/TYgg5p8Xat9ckLbfFz7/afBLce1Lk7lgs5KKBTP7aRE2/wgATx+o+7BBADtjU/HmFkU6Zmu9ExXpp4lIhLUZGel9sFxbfMxbemXwa+qtCE5ykX5yszelmG5Z8p4VaZPAEBLCMK2sigyKFsZlK3ccLaISJ1fPizT3jgUnPHfgM0i1/SxXNNb6ZeZ4BDyMFgGAFpAEMaZS5UL85UL862/HS6fVWjLdwfHvRrItitX91Zm97P0an5/gHbnCUgqQQgAzTnzurOSyKBs5aHzrQevsT051lrhlZH/9o99xf/ojuBxT0eXhFGjANASgrDdWRQZm688Osr69Szb7d+xbDqmDXip4fGdwTis6KObN3ByW3MAQBMEYcdJscr3z7KsvMT6wffVF/cEL1rj313dQWlI0ygAtIQgTIDe6cr6Sep1fS0XvuJ/aldHVA09AY3BMgDQLIIwMSyK3FhkeWuS+tcvghP/4z9Q175p6A0I0ycAoFkEYSINzFI2TVEvKbAMX+1/alew/f4h1hoFgJYQhAmmWuSucyzrJ6lP7Qpe8br/YPtUDRk1CgAtIQjPCIOyleKp6sVdLcNfbpeqIUEIAC1hQv2ZIlQ1vKxQ+cFbgbPSlAnd4tmlR9MoALSEGuGZZViu8seRljs/CMR3LCk1QgBoCUF4xvn+WZZOKbJ8dzwbSFl0GwBaQhCeiR4aYb13a9Dtj9sFqRECQEsIwjPRiM7KyDxlyWdxqxTSRwgALSEIz1APj7D8YUeg3Bufq1EjBICWEIRnqN7pyoxelkXbA3G5GkEIAC0hCM9cvx5mXb47uCceC3PTNAoALSEIz1ydHbJgkPVX2+LQU0iNEABaQhCe0X45xPJuqVZ8rK2VQqZPAEBLCMIzWqoqvx5muXtLW3sKaRoFgJYQhGe6G862lHvk1QNtqhTSNAoALSEIz3RWRf7vfOsdmwP+1vYVBjQJaGLjTw0AzeHT0QAm91AKnLLsq1YmoZfqIAC0jCA0hv8733rfh8G6Vi26RrsoAERAEBrD+Z2VC/OVP+1oTaWQIaMAEAFBaBgPnmf5047A0fqYf5AhowAQAUFoGL3Slf/pa1n0UcxTKWgaBYAICEIjuXeo9cU9MS+6Ro0QACIgCI2kU4qM62rZVhZbEFIjBIAICEKDcakS64a9BCEAREAQGozLJrFOoiAIASACgtBgWlEj9Aa0FKZPAEALCEKDcapS56ePEADihiA0GJeq1DXE9iMEIQBEQBAajEttTR8h0ycAoCUEocG4bK3oI6RGCAAtIggNxtmqGiFBCAAtIQgNxqUqrRgsQ9MoALSEIDQYlyqxDpZh+gQAREAQGgx9hAAQXwShwbRu1ChBCAAtIQgNhsEyABBfBKHBuFSlroFtmAAgbghCg3Gp4o5xa15qhAAQAUFoMA5VfAEJxFIn9AQ0B6NGAaAFBKHBKCKpqtTH0k1I0ygAREAQGk+s42VoGgWACAhC44l1cRmCEAAiIAiNJ9bFZVhiDQAiIAiNJ9bFZVhZBgAiIAiNx2mljxAA4oYgNB6XTWLsI2TRbQBoEUFoPC5ViamPkKZRAIiAIDSeWBeXoWkUACIgCI3HZYth1Kgm0hAUO0EIAC0gCI0npgn13oDYLUIPIQC0hCA0HpequHUPlqFdFAAiIwiNJ6a9eVloFAAiIwiNxxnLyjJsPQEAkRGExuOyxVAjpGkUACIjCI0npqZRFhoFgMgIQuOJabAMs+kBIDKC0Hhimj5B0ygAREYQGk9M2zDRNAoAkRGExhPTYBkvo0YBICK17Zfw+Xwvvvji4cOHx44dO3bs2CZHX3/99Y8++qh3797Tp0+3WCyBQGDjxo0ff/yxqqqXX355//79RWTdunUVFRWh83NycsaPH9/2UiUxlxrDfoQ0jQJAZHGoEU6aNGnp0qUej2fmzJlPP/1040MPPPDALbfc4vP5Fi9ePGfOnNArd9xxx6FDh3bu3HneeeetWrVKRO66665ly5atW7du3bp1W7ZsaXuRkptLVfRvw0QQAkAUWtu8/fbbXbp0cbvdmqatXbu2e/fufr8/dKi6ujo9Pf2jjz7SNK28vNzpdH755ZcnTpwI/+zixYtHjhypadqwYcPefPPNqP/W8ePHO3XqpL9s1dXVMd2LUXj8mv0ZX4QTGt/407sCP3rb3/6FOiN4vV6v15voUiRGsr7bozLtjbvd7oaGhkSXIgGCwWBtbW18r9nWGuGbb745fvz41NRUEbnkkkvKysp27doVOrR58+bMzMxzzjlHRHJyckaNGrVu3brs7Ozwz6akpKjqybbZjRs3Pvfcc9u2bWtjecwgxSpBkYagrpOpEQJAZG3tIzxy5Eh+fv7Ja6lqbm7uoUOHBg0aJCKHDx8OHxKR/Pz8w4cPh5+WlZU9/PDDf/rTn0SkR48eJSUlBw8evP3226dPn/7EE0+09M/5fL477rij8Ss9evS48cYbmz3Z6/Xa7fY23NyZy2m1VNR5M1u4ucY3XutVbCJer+5ORSPz+Xwioml6242TSRK/2yMz840Hg8FAIJa9SZOCpmlerzdciYrKbrcrSpQBg3qvNXLkyD179jR58cYbb1QUpfHnjqZpVuvJCojFYmnpUE1NzZQpU66++urp06eLyOrVq0OvHzhwYMCAAXPmzBk5cmSzxVAUJSsrq/Er6enpFkvz9VqLxdLSIaNz2cQTtGS3cHONb9wXFIcqFospBo6G7jpZ/+iRJfG7PTKT37gJ713TtLjfuN4gfOONN07/6uFwOB5++OEvvvgi9LShoaGsrKxr166hpwUFBUeOHAmffOTIkQsvvFBE6urqJk+efM455/zhD39ocsHu3bsPGDBg165dLQWhzWZbuHChzjLbbDabzabzZGNxqX6fYrXZmo+3xjful0CqTbHZTPG/JfTFK1n/6JEl8bs9MtPeuN/vt9ls+itGSUPTtLj/0fV+PmZkZGSfJjU1deLEiW+99VZdXZ2IvPHGG/n5+UVFReXl5V999dXIkSPdbvfWrVtF5Pjx48XFxZdffnl9ff3UqVN79er1l7/8JVRdDQa/7ew6cuTIF1980a9fvzjeYVLSv7gMfYQAEFlbv02MHDly1KhRl1xyyUUXXbR8+fLFixdbLJaVK1c+/fTT27dvX7hw4bRp06655prXXnvt+uuv79279//+7/++8847U6dOnTVrlohkZmYuWLDg2muvHTNmjKIoq1evnjZt2pgxY+Jxa8lM/+IyBCEARBaHavU///nPV1555cCBA6+//vq5554rIqGWTxG5/fbbR48evW3btsWLF0+YMEFEZs+efckll4R/1m639+/ff8mSJZ9//rnVap0zZ86IESPaXqSkp39xGZZYA4DI4hCEVqv1+9//fuNXevTo0aNHj9Dj0aNHjx49Onyof//+odVkGhs3bty4cePaXhLz+GYDiuhDYNh9AgAiM8UYiuRDHyEAxAtBaEgx9RHSNAoAERCEhqS/j5DdJwAgMoLQkPRvQEHTKABERhAaklP3BhQEIQBERhAakkv3YBkvfYQAEBFBaEg0jQJAvBCEhuSysbIMAMQHQWhI+ucRegNaCqNGAaBlBKEhuRgsAwBxQhAaEotuA0C8EISG5LLpHSzDqFEAiIwgNCSnVVcfoS8oqiLm2J0eAFqJIDQkl0309BGy0CgAREUQGpJLVfT0EbIHEwBERRAakksVdyD6aR5W3AaAaAhCQ1ItYhHxBaOc5vGLIw5bLwNAMiMIjUrP4jLeoKTwFwaAiPiYNCo9G1AwiRAAoiIIjUrPBhQ0jQJAVAShUenZgIKmUQCIio9Jo9LTR0jTKABERRAalZ7FZTx+zaEyfQIAIiEIjcpl0zVYhqZRAIiMj0mj0tlHSNMoAERGEBoVo0YBIC4IQqNy6tiSkKZRAIiKj0mjctmi1whpGgWAqAhCo3KpijvaYBlvQEth0W0AiIggNCpdfYTMIwSAaAhCo9LZR0gQAkBkBKFR6akRetmhHgCiIQiNymWLPo+QGiEAREUQGpWLbZgAIB4IQqNy0jQKAPFAEBqVS9dgGc3B9AkAiIggNCpdfYQssQYA0RCERqWzj5Al1gAgMj4mjUpXHyFLrAFANAShUblUqfdL5CohTaMAEBVBaFQWRWwW8USsFNI0CgBR8TFpYFEXl6FpFACiIggNzGWLsgGFx685VKZPAEAkBKGBRa0RsrIMAERFEBpY1IGj3iB9hAAQBR+TBhZ5cRl/UDRNVP7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       "  481.399,77.0856 491.598,77.1307 501.798,77.1731 511.998,77.2131 522.197,77.2509 532.397,77.2866 542.597,77.3205 552.796,77.3525 562.996,77.383 573.196,77.412 \n",
       "  \n",
       "  \"/>\n",
       "<path clip-path=\"url(#clip400)\" d=\"\n",
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot(saving, temp_1[3] .- 0.02/4)\n",
    "plot(saving, temp_2[3] - temp_1[3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "50-element Vector{Float64}:\n",
       "  0.003716541300945453\n",
       "  0.0035374540406123423\n",
       "  0.003300284425576283\n",
       "  0.002971307217623176\n",
       "  0.0024844209498531278\n",
       "  0.0016900275655959973\n",
       "  0.00016234798048664345\n",
       " -0.0039842108933818565\n",
       " -0.0578894762536718\n",
       "  0.017577895250734166\n",
       "  ⋮\n",
       "  0.005319235920069332\n",
       "  0.0053098003756337775\n",
       "  0.005300906584945642\n",
       "  0.005292509191877404\n",
       "  0.0052845677658535095\n",
       "  0.00527704615089708\n",
       "  0.005269911915251724\n",
       "  0.005263135883906367\n",
       "  0.005256691739810915"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "temp_1[3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "PortfolioChoice (generic function with 1 method)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "function PortfolioChoice(saving ;b_e_bar=0.21, ϕ=0.05, r_d=0.02/4, r_e=0.02/4)\n",
    "    #= \n",
    "    Max: b_d + b_e- Φ(b_e)\n",
    "    s.t.: q_d*b_d + q_e*b_e = b\n",
    "    =>\n",
    "    Φ′(b_e) = 1-q_e/q_d\n",
    "    under the assumption Φ(b_e)=ϕ/2*(b_e - b_e_bar)^2 => b_e = b_e_bar + 1/ϕ*(r_e-r_d)/(1+r_e)\n",
    "    =#\n",
    "    q_d = 1/(1+r_d); q_e = 1/(1+r_e);\n",
    "    b_e = (r_e-r_d)/ϕ/(1+r_e) + b_e_bar; b_e = fill(b_e, size(saving));\n",
    "    b_d = (saving .- q_e*b_e )/q_d;\n",
    "    b = b_d + b_e- ϕ/2*(b_e .- b_e_bar).^2;\n",
    "    EffRet = b ./ saving .- 1;\n",
    "    return b_e, b_d, EffRet, b\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "saving = range(-1, 5, length=50);\n",
    "temp_1 = PortfolioChoice(saving; r_e=(0.02-0.02)/4);\n",
    "temp_2 = PortfolioChoice(saving; r_e=(0.02+0.04)/4);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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   ],
   "source": [
    "plot(saving, temp_1[3] .- 0.02/4)"
   ]
  },
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   "metadata": {},
   "source": [
    "Note:\n",
    "\n",
    "1. The effective return rate difference depends on the level of saving: for the very high saving rate, since most of the saving is still in domestic bond, the effective return rate will be very close to the domestic bond return rate;\n",
    "2. When someone is borrowing, \n",
    "    2.1. if r_e>r_d, the ideal choice is to borrow domestic bond as much as it can, and save in foreign bond"
   ]
  }
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